Subjective expected utility

Subjective expected utility is a method in decision theory in the presence of risk, promoted by L. J. Savage in 1954 following previous work by Ramsey and von Neumann. The theory of subjective expected utility combines two subjective concepts: first, a personal utility function, and second a personal probability distribution (based on Bayesian probability theory).

Savage proved that, if you adhere to axioms of rationality, if you believe an uncertain event has possible outcomes $\{x_i\}$ each with a utility to you of $u(x_i)$ then your choices can be explained as arising from a function in which you believe that there is a subjective probability of each outcome is $P(x_i)$, and your subjective expected utility is the expected value of the utility,

$\sum_i \; u(x_i) \; P(x_i).$

You may be able to make a decision which changes the possible outcomes to $\{y_j\}$ in which case your subjective expected utility will become

$\sum_j \; u(y_j) \; P(y_j).$

Which decision you prefer depends on which subjective expected utility is higher. Different people may make different decisions because they may have different utility functions or different beliefs about the probabilities of different outcomes.

Savage assumed that it is possible to take convex combinations of decisions and that preferences would be preserved. So if you prefer $x(=\{x_i\})$ to $y$ and $s$ to $t$ then you will prefer $\lambda x + (1-\lambda )s$ to $\lambda y + (1-\lambda )t$, for $0<\lambda<1$.

Experiments have shown that many individuals do not behave in a manner consistent with subjective expected utility, e.g. most prominently Allais (1953) and Ellsberg (1961) .

References

• Charles Sanders Peirce and Joseph Jastrow (1885). "On Small Differences in Sensation". Memoirs of the National Academy of Sciences 3: 73–83.  http://psychclassics.yorku.ca/Peirce/small-diffs.htm
• Ramsey, Frank Plumpton; “Truth and Probability” (PDF), Chapter VII in The Foundations of Mathematics and other Logical Essays (1931).
• de Finetti, Bruno. "Probabilism: A Critical Essay on the Theory of Probability and on the Value of Science," (translation of 1931 article) in Erkenntnis, volume 31, September 1989.
• de Finetti, Bruno. 1937, “La Prévision: ses lois logiques, ses sources subjectives,” Annales de l'Institut Henri Poincaré,

de Finetti, Bruno. "Foresight: its Logical Laws, Its Subjective Sources," (translation of the 1937 article in French) in H. E. Kyburg and H. E. Smokler (eds), Studies in Subjective Probability, New York: Wiley, 1964.

• de Finetti, Bruno. Theory of Probability, (translation by AFM Smith of 1970 book) 2 volumes, New York: Wiley, 1974–5.
• Donald Davidson, Patrick Suppes and Sidney Siegel (1957). Decision-Making: An Experimental Approach. Stanford University Press. 
• Pfanzagl, J (1967). "Subjective Probability Derived from the Morgenstern–von Neumann Utility Theory". In Martin Shubik. Essays in Mathematical Economics In Honor of Oskar Morgenstern. Princeton University Press. pp. 237–251. 
• Pfanzagl, J. (1968). "Events, Utility and Subjective Probability". Theory of Measurement. Wiley. pp. 195–220. 
• Morgenstern, Oskar (1976). "Some Reflections on Utility". In Andrew Schotter. Selected Economic Writings of Oskar Morgenstern. New York University Press. pp. 65–70. ISBN 0-8147-7771-6. 

External links

• Edi Karni, Johns Hopkins University (November 9, 2005). "Savages’ Subjective Expected Utility Model". Retrieved 2009-02-17.